Which of the following is a subspace of R3?

Which of the following is a subspace of R3?

If you did not yet know that subspaces of R3 include: the origin (0-dimensional), all lines passing through the origin (1-dimensional), all planes passing through the origin (2-dimensional), and the space itself (3-dimensional), you can still verify that (a) and (c) are subspaces using the Subspace Test.

Is X Y Z 2 a subspace of R3?

No, it is not a subspace of R³ . Clearly u = (1, 0, -1) , v = (1, 0, 1) both belong to W but (u+ v) = (2, 0, 0) does not belong to W because, (4 + 0) = 4 ≉ 0 etc.

Which one of the following is not a subspace of R 3?

The plane z = 1 is not a subspace of R3. The line t(1,1,0), t ∈ R is a subspace of R3 and a subspace of the plane z = 0. The line (1,1,1) + t(1,−1,0), t ∈ R is not a subspace of R3 as it lies in the plane x + y + z = 3, which does not contain 0. Any solution (x1,x2,…,xn) is an element of Rn.

Is R3 a subspace of R3?

And R3 is a subspace of itself. Every line through the origin is a subspace of R3 for the same reason that lines through the origin were subspaces of R2. The other subspaces of R3 are the planes pass- ing through the origin. Let W be a plane passing through 0.

How many subspaces does R 3 have?

In R3, there are zero, 1, 2, 3 dimensional subspaces.

Is R 2 a subspace of R 3?

Instead, most things we want to study actually turn out to be a subspace of something we already know to be a vector space. However, R2 is not a subspace of R3, since the elements of R2 have exactly two entries, while the elements of R3 have exactly three entries. That is to say, R2 is not a subset of R3.

Is WA subspace of R3?

Therefore by Theorem 4.2 W is a subspace of R3. (e) (1,1,0) ∈ W, however 2(1,1,0) = (2,2,0) ∈ W, so W is not closed under scalar multiplication and so it is not a subspace of R3.

Is R2 subset of R3?

How do you determine if a set is a subspace of R 3?

A subset of R3 is a subspace if it is closed under addition and scalar multiplication. Besides, a subspace must not be empty. The set S1 is the union of three planes x = 0, y = 0, and z = 0. It is not closed under addition as the following example shows: (1,1,0) + (0,0,1) = (1,1,1).

What is a subset of R3?

A subset of R3 is a subspace if it is closed under addition and scalar multiplication. Besides, a subspace must not be empty. The set S1 is the union of three planes x = 0, y = 0, and z = 0.

Is R2 part of R3?

Things in R^2 are of the form (a, b), with two components while things in R^3 are of the form (a, b, c) with three components. Members of R^2 are not members of R^3 so R^2 is not a subset of R^3.

How do you find the subspace of R3?

In other words, to test if a set is a subspace of a Vector Space, you only need to check if it closed under addition and scalar multiplication. Easy! ex. Test whether or not the plane 2x + 4y + 3z = 0 is a subspace of R3.

Is W a subspace of R3?

Therefore, the only vector in W is the zero vector. So, W is nonempty and a subset of R 3. Furthermore, because W is closed under addition and scalar multiplication, it is a subspace of R 3. where x = a 1 + b 1, y = a 2 + b 2, z = a 3 + b 3. = (x,y,z) Closure under addition.

Is W nonempty and a subset of R3?

3. Since, x + y + z = 0. Then, the values for all the variables have to be zero. Therefore, the only vector in W is the zero vector. So, W is nonempty and a subset of R 3. Furthermore, because W is closed under addition and scalar multiplication, it is a subspace of R 3.

When is a subset of R3 closed under scalar multiplication?

A subset S of R 3 is closed under scalar multiplication if any real multiple of any vector in S is also in S. In other words, if r is any real number and ( x 1, y 1, z 1) is in the subspace, then so is ( r x 1, r y 1, r z 1).

What are the different types of subsets in math?

Subsets are classified as. Proper Subset. Improper Subsets. A proper subset is one that contains few elements of the original set whereas an improper subset, contains every element of the original set along with the null set.